Table of Perfect Lattices

Keywords: tables, perfect lattices, quadratic forms

Part of the Catalogue of Lattices which is a joint project of Gabriele Nebe, RWTH Aaachen (nebe@math.rwth-aachen.de) and Neil J. A. Sloane, AT&T Labs-Research (njas@research.att.com).

Last modified Nov 14 2005

Perfect Lattices

This table of perfect lattices is complete through 7 dimensions.

• 1 dimension P1.1 = A1

• 2 dimensions P2.1 = A2

• 3 dimensions P3.1 = A3

• 4 dimensions P4.1 = D4, P4.2 = A4

• 5 dimensions P5.1 = D5, P5.2 = (A5)+3, P5.3 = A5

• 6 dimensions P6.1 = E6, P6.2 = E6*, P6.3 = D6, P6.4 = A6,2, P6.5 = A6 sup (2), P6.6 = A6,1, P6.7 = A6

• 7 dimensions P7.1 = E7, P7.2 = E7*, P7.3, P7.4 = D7, P7.5, P7.6, P7.7, P7.8, P7.9, P7.10, P7.11, P7.12, P7.13, P7.14, P7.15, P7.16, P7.17, P7.18, P7.19, P7.20, P7.21, P7.22, P7.23, P7.24, P7.25, P7.26, P7.27, P7.28, P7.29, P7.30, P7.31, P7.32, P7.33 = A7.

Remarks about the enumeration in 8 dimensions

The numbers of perfect lattices are given in sequence A004026 in the On-Line Encyclopedia of Integer Sequences.

For additional information see the home page of Jacques Martinet.

In 2001, Professor Martinet (martinet@math.u-bordeaux.fr) reported as follows.

The complete classification of 8-dimensional perfect lattices is not known. But we have some material that I now describe, referring to my book (see Reference 3 below), chapter VI, section 6.

1. Th. 6.5 : there are 1175 perfect 8-dimensional lattices having a perfect 7-dimensional section with the same norm. (This was essentially done by Mohamed La{\"\i}hem (written now Laihem), an Algerian student of Anne-Marie Berg\'e, who classified the 1171 lattices attached to 7-dimensional perfect sections other that E_7, D_7, A_7, a result then completed by another student of Prof. Berge' (Jean-Luc Baril); the four special lattices are E_8, Barnes' A_8^2, D_8 and A_8.

This data can be obtained via PARI-gp : Batut has written a small PARI program which combines a 7-dimensional perfect Gram matrix from your paper with Conway and the last column of a symmetric 8-dimensional matrix.

2. A student of mine (Huguette Napias; see p. 183 in my book) has found altogether 10170 8-dimensional perfect lattices. She explored the Vorono{\"\i} graph, starting with Laihem's and Baril's lists, then listing all new neighbours of those with s=8(8+1)/2=36, then all new neighbours of the new lattices, and so on, until the process stops.

Her list must be "almost" but not quite complete, for one cannot exclude the existence of a few lattices connected only to E_8 (or A_8^2) which would have been invisible, as would have been E_7^* without a complete exploration of E_7. As far as I remember, E_8 possesses over 6000 inequivalent Vorono{\"\i} facets, so that a systematic exploration of the graph is not possible.

3. As for a web site, I hope to make one soon. The problem is the use of PARI-gp for Laihem's list and, whatever we do, the very long list of Napias with more than 9000 entries.

References

• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices III: Perfect Forms, Proc. Royal Soc. London, Series A, volume 418, pages 43-80, 1988.
• D.-O. Jaquet-Chiffelle, Enume'ration comple`te des classes de formes parfaites en dimension 7, Ann. Inst. Fourier, 43 (1993), 21-55, showed that the known list of perfect 7-dim lattices is indeed complete.
• J. Martinet, Les Re'seaux Parfaits des Espaces Euclidiens, Masson, Paris, 1996.
• J. Martinet, Home page (among other things, lists all known 8-dimensional perfect lattices).

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